A weighted quadratic ratio lower bound at fixed sum three

Proved

WorkbookSource.base_18533

by wamlart · Sep 13, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

Let $a,b,c> 0$ and $a+b+c=3$ . prove that: $\frac{a^2}{9a+(b+2c)^2}+\frac{b^2}{9b+(c+2a)^2}+\frac{c^2}{9c+(a+2b)^2}\geq \frac{1}{6}$ .

Source: InternLM Lean-Workbook, record lean_workbook_18533 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_18533 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : (a^2 / (9 * a + (b + 2 * c)^2) + b^2 / (9 * b + (c + 2 * a)^2) + c^2 / (9 * c + (a + 2 * b)^2)) ≥ 1 / 6  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_18533; Apache-2.0

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